LMFDB - Embedded newform 4851.2.a.bp.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4851,2,Mod(1,4851)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4851, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4851.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4851 = 3^{2} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4851.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$38.7354300205$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.837.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 1 $ x^3 - 6*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.52892$ of defining polynomial
Character	$\chi$	$=$	4851.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.52892 q^{2} +4.39543 q^{4} +0.133492 q^{5} -6.05784 q^{8} +O(q^{10})$	$q-2.52892 q^{2} +4.39543 q^{4} +0.133492 q^{5} -6.05784 q^{8} -0.337590 q^{10} -1.00000 q^{11} -0.133492 q^{13} +6.52892 q^{16} -5.05784 q^{17} +0.924344 q^{19} +0.586754 q^{20} +2.52892 q^{22} +7.05784 q^{23} -4.98218 q^{25} +0.337590 q^{26} -3.86651 q^{29} -2.79085 q^{31} -4.39543 q^{32} +12.7909 q^{34} +9.98218 q^{37} -2.33759 q^{38} -0.808672 q^{40} +11.8487 q^{41} -3.05784 q^{43} -4.39543 q^{44} -17.8487 q^{46} -3.07566 q^{47} +12.5995 q^{50} -0.586754 q^{52} +4.79085 q^{53} -0.133492 q^{55} +9.77808 q^{58} -12.6574 q^{59} -6.00000 q^{61} +7.05784 q^{62} -1.94216 q^{64} -0.0178201 q^{65} +8.92434 q^{67} -22.2313 q^{68} +6.11567 q^{71} -7.86651 q^{73} -25.2441 q^{74} +4.06289 q^{76} +14.1157 q^{79} +0.871558 q^{80} -29.9644 q^{82} -1.20915 q^{83} -0.675180 q^{85} +7.73302 q^{86} +6.05784 q^{88} +15.5817 q^{89} +31.0222 q^{92} +7.77808 q^{94} +0.123392 q^{95} -12.7909 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{4} - 3 q^{8}+O(q^{10}) $ 3 * q + 6 * q^4 - 3 * q^8	$ 3 q + 6 q^{4} - 3 q^{8} - 9 q^{10} - 3 q^{11} + 12 q^{16} - 12 q^{19} - 21 q^{20} + 6 q^{23} + 15 q^{25} + 9 q^{26} - 12 q^{29} + 6 q^{31} - 6 q^{32} + 24 q^{34} - 15 q^{38} - 18 q^{40} + 6 q^{41} + 6 q^{43} - 6 q^{44} - 24 q^{46} - 24 q^{47} + 39 q^{50} + 21 q^{52} - 9 q^{58} - 24 q^{59} - 18 q^{61} + 6 q^{62} - 21 q^{64} - 30 q^{65} + 12 q^{67} - 6 q^{68} - 12 q^{71} - 24 q^{73} - 39 q^{74} + 3 q^{76} + 12 q^{79} + 9 q^{80} - 30 q^{82} - 18 q^{83} - 18 q^{85} + 24 q^{86} + 3 q^{88} + 18 q^{89} + 18 q^{92} - 15 q^{94} - 12 q^{95} - 24 q^{97}+O(q^{100}) $ 3 * q + 6 * q^4 - 3 * q^8 - 9 * q^10 - 3 * q^11 + 12 * q^16 - 12 * q^19 - 21 * q^20 + 6 * q^23 + 15 * q^25 + 9 * q^26 - 12 * q^29 + 6 * q^31 - 6 * q^32 + 24 * q^34 - 15 * q^38 - 18 * q^40 + 6 * q^41 + 6 * q^43 - 6 * q^44 - 24 * q^46 - 24 * q^47 + 39 * q^50 + 21 * q^52 - 9 * q^58 - 24 * q^59 - 18 * q^61 + 6 * q^62 - 21 * q^64 - 30 * q^65 + 12 * q^67 - 6 * q^68 - 12 * q^71 - 24 * q^73 - 39 * q^74 + 3 * q^76 + 12 * q^79 + 9 * q^80 - 30 * q^82 - 18 * q^83 - 18 * q^85 + 24 * q^86 + 3 * q^88 + 18 * q^89 + 18 * q^92 - 15 * q^94 - 12 * q^95 - 24 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.52892	−1.78822	−0.894108	−	0.447852i	$-0.852189\pi$
$2$	−2.52892	−1.78822	−0.894108	+	0.447852i	$0.852189\pi$
$3$	0	0
$3$	0	0
$4$	4.39543	2.19771
$5$	0.133492	0.0596994	0.0298497	−	0.999554i	$-0.490497\pi$
$5$	0.133492	0.0596994	0.0298497	+	0.999554i	$0.490497\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−6.05784	−2.14177
$9$	0	0
$10$	−0.337590	−0.106755
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−0.133492	−0.0370240	−0.0185120	−	0.999829i	$-0.505893\pi$
$13$	−0.133492	−0.0370240	−0.0185120	+	0.999829i	$0.505893\pi$
$14$	0	0
$15$	0	0
$16$	6.52892	1.63223
$17$	−5.05784	−1.22671	−0.613353	−	0.789809i	$-0.710180\pi$
$17$	−5.05784	−1.22671	−0.613353	+	0.789809i	$0.710180\pi$
$18$	0	0
$19$	0.924344	0.212059	0.106030	−	0.994363i	$-0.466186\pi$
$19$	0.924344	0.212059	0.106030	+	0.994363i	$0.466186\pi$
$20$	0.586754	0.131202
$21$	0	0
$22$	2.52892	0.539167
$23$	7.05784	1.47166	0.735830	−	0.677166i	$-0.236792\pi$
$23$	7.05784	1.47166	0.735830	+	0.677166i	$0.236792\pi$
$24$	0	0
$25$	−4.98218	−0.996436
$26$	0.337590	0.0662069
$27$	0	0
$28$	0	0
$29$	−3.86651	−0.717993	−0.358996	−	0.933339i	$-0.616881\pi$
$29$	−3.86651	−0.717993	−0.358996	+	0.933339i	$0.616881\pi$
$30$	0	0
$31$	−2.79085	−0.501252	−0.250626	−	0.968084i	$-0.580636\pi$
$31$	−2.79085	−0.501252	−0.250626	+	0.968084i	$0.580636\pi$
$32$	−4.39543	−0.777009
$33$	0	0
$34$	12.7909	2.19361
$35$	0	0
$36$	0	0
$37$	9.98218	1.64106	0.820530	−	0.571603i	$-0.193678\pi$
$37$	9.98218	1.64106	0.820530	+	0.571603i	$0.193678\pi$
$38$	−2.33759	−0.379207
$39$	0	0
$40$	−0.808672	−0.127862
$41$	11.8487	1.85045	0.925227	−	0.379414i	$-0.123874\pi$
$41$	11.8487	1.85045	0.925227	+	0.379414i	$0.123874\pi$
$42$	0	0
$43$	−3.05784	−0.466316	−0.233158	−	0.972439i	$-0.574906\pi$
$43$	−3.05784	−0.466316	−0.233158	+	0.972439i	$0.574906\pi$
$44$	−4.39543	−0.662635
$45$	0	0
$46$	−17.8487	−2.63165
$47$	−3.07566	−0.448631	−0.224315	−	0.974517i	$-0.572015\pi$
$47$	−3.07566	−0.448631	−0.224315	+	0.974517i	$0.572015\pi$
$48$	0	0
$49$	0	0
$50$	12.5995	1.78184
$51$	0	0
$52$	−0.586754	−0.0813681
$53$	4.79085	0.658074	0.329037	−	0.944317i	$-0.393276\pi$
$53$	4.79085	0.658074	0.329037	+	0.944317i	$0.393276\pi$
$54$	0	0
$55$	−0.133492	−0.0180000
$56$	0	0
$57$	0	0
$58$	9.77808	1.28393
$59$	−12.6574	−1.64785	−0.823924	−	0.566700i	$-0.808220\pi$
$59$	−12.6574	−1.64785	−0.823924	+	0.566700i	$0.808220\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	7.05784	0.896346
$63$	0	0
$64$	−1.94216	−0.242771
$65$	−0.0178201	−0.00221031
$66$	0	0
$67$	8.92434	1.09028	0.545141	−	0.838344i	$-0.316476\pi$
$67$	8.92434	1.09028	0.545141	+	0.838344i	$0.316476\pi$
$68$	−22.2313	−2.69595
$69$	0	0
$70$	0	0
$71$	6.11567	0.725797	0.362898	−	0.931829i	$-0.381787\pi$
$71$	6.11567	0.725797	0.362898	+	0.931829i	$0.381787\pi$
$72$	0	0
$73$	−7.86651	−0.920705	−0.460353	−	0.887736i	$-0.652277\pi$
$73$	−7.86651	−0.920705	−0.460353	+	0.887736i	$0.652277\pi$
$74$	−25.2441	−2.93457
$75$	0	0
$76$	4.06289	0.466045
$77$	0	0
$78$	0	0
$79$	14.1157	1.58814	0.794069	−	0.607828i	$-0.207959\pi$
$79$	14.1157	1.58814	0.794069	+	0.607828i	$0.207959\pi$
$80$	0.871558	0.0974431
$81$	0	0
$82$	−29.9644	−3.30901
$83$	−1.20915	−0.132721	−0.0663606	−	0.997796i	$-0.521139\pi$
$83$	−1.20915	−0.132721	−0.0663606	+	0.997796i	$0.521139\pi$
$84$	0	0
$85$	−0.675180	−0.0732336
$86$	7.73302	0.833873
$87$	0	0
$88$	6.05784	0.645767
$89$	15.5817	1.65166	0.825829	−	0.563921i	$-0.190708\pi$
$89$	15.5817	1.65166	0.825829	+	0.563921i	$0.190708\pi$
$90$	0	0
$91$	0	0
$92$	31.0222	3.23429
$93$	0	0
$94$	7.77808	0.802248
$95$	0.123392	0.0126598
$96$	0	0
$97$	−12.7909	−1.29871	−0.649357	−	0.760484i	$-0.724962\pi$
$97$	−12.7909	−1.29871	−0.649357	+	0.760484i	$0.724962\pi$
$98$	0	0
$99$	0	0
$100$	−21.8988	−2.18988
$101$	−9.59180	−0.954420	−0.477210	−	0.878789i	$-0.658352\pi$
$101$	−9.59180	−0.954420	−0.477210	+	0.878789i	$0.658352\pi$
$102$	0	0
$103$	−9.84869	−0.970420	−0.485210	−	0.874398i	$-0.661257\pi$
$103$	−9.84869	−0.970420	−0.485210	+	0.874398i	$0.661257\pi$
$104$	0.808672	0.0792968
$105$	0	0
$106$	−12.1157	−1.17678
$107$	−0.924344	−0.0893597	−0.0446799	−	0.999001i	$-0.514227\pi$
$107$	−0.924344	−0.0893597	−0.0446799	+	0.999001i	$0.514227\pi$
$108$	0	0
$109$	−8.52387	−0.816439	−0.408219	−	0.912884i	$-0.633850\pi$
$109$	−8.52387	−0.816439	−0.408219	+	0.912884i	$0.633850\pi$
$110$	0.337590	0.0321880
$111$	0	0
$112$	0	0
$113$	12.1157	1.13975	0.569873	−	0.821733i	$-0.306992\pi$
$113$	12.1157	1.13975	0.569873	+	0.821733i	$0.306992\pi$
$114$	0	0
$115$	0.942164	0.0878573
$116$	−16.9950	−1.57794
$117$	0	0
$118$	32.0094	2.94671
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	15.1735	1.37374
$123$	0	0
$124$	−12.2670	−1.10161
$125$	−1.33254	−0.119186
$126$	0	0
$127$	−8.90652	−0.790326	−0.395163	−	0.918611i	$-0.629312\pi$
$127$	−8.90652	−0.790326	−0.395163	+	0.918611i	$0.629312\pi$
$128$	13.7024	1.21113
$129$	0	0
$130$	0.0450656	0.00395251
$131$	−15.6974	−1.37149	−0.685743	−	0.727844i	$-0.740523\pi$
$131$	−15.6974	−1.37149	−0.685743	+	0.727844i	$0.740523\pi$
$132$	0	0
$133$	0	0
$134$	−22.5689	−1.94966
$135$	0	0
$136$	30.6395	2.62732
$137$	−14.6395	−1.25074	−0.625370	−	0.780328i	$-0.715052\pi$
$137$	−14.6395	−1.25074	−0.625370	+	0.780328i	$0.715052\pi$
$138$	0	0
$139$	−18.6496	−1.58184	−0.790921	−	0.611918i	$-0.790398\pi$
$139$	−18.6496	−1.58184	−0.790921	+	0.611918i	$0.790398\pi$
$140$	0	0
$141$	0	0
$142$	−15.4660	−1.29788
$143$	0.133492	0.0111632
$144$	0	0
$145$	−0.516148	−0.0428637
$146$	19.8938	1.64642
$147$	0	0
$148$	43.8759	3.60658
$149$	−11.8665	−0.972142	−0.486071	−	0.873919i	$-0.661570\pi$
$149$	−11.8665	−0.972142	−0.486071	+	0.873919i	$0.661570\pi$
$150$	0	0
$151$	−11.3248	−0.921601	−0.460800	−	0.887504i	$-0.652438\pi$
$151$	−11.3248	−0.921601	−0.460800	+	0.887504i	$0.652438\pi$
$152$	−5.59952	−0.454181
$153$	0	0
$154$	0	0
$155$	−0.372556	−0.0299244
$156$	0	0
$157$	20.3827	1.62671	0.813357	−	0.581766i	$-0.197638\pi$
$157$	20.3827	1.62671	0.813357	+	0.581766i	$0.197638\pi$
$158$	−35.6974	−2.83993
$159$	0	0
$160$	−0.586754	−0.0463870
$161$	0	0
$162$	0	0
$163$	−19.3070	−1.51224	−0.756120	−	0.654432i	$-0.772908\pi$
$163$	−19.3070	−1.51224	−0.756120	+	0.654432i	$0.772908\pi$
$164$	52.0800	4.06677
$165$	0	0
$166$	3.05784	0.237334
$167$	12.6395	0.978077	0.489038	−	0.872262i	$-0.337348\pi$
$167$	12.6395	0.978077	0.489038	+	0.872262i	$0.337348\pi$
$168$	0	0
$169$	−12.9822	−0.998629
$170$	1.70748	0.130957
$171$	0	0
$172$	−13.4405	−1.02483
$173$	−19.8487	−1.50907	−0.754534	−	0.656261i	$-0.772137\pi$
$173$	−19.8487	−1.50907	−0.754534	+	0.656261i	$0.772137\pi$
$174$	0	0
$175$	0	0
$176$	−6.52892	−0.492136
$177$	0	0
$178$	−39.4049	−2.95352
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−2.67518	−0.198845	−0.0994223	−	0.995045i	$-0.531699\pi$
$181$	−2.67518	−0.198845	−0.0994223	+	0.995045i	$0.531699\pi$
$182$	0	0
$183$	0	0
$184$	−42.7552	−3.15196
$185$	1.33254	0.0979703
$186$	0	0
$187$	5.05784	0.369866
$188$	−13.5188	−0.985961
$189$	0	0
$190$	−0.312049	−0.0226384
$191$	−3.32482	−0.240576	−0.120288	−	0.992739i	$-0.538382\pi$
$191$	−3.32482	−0.240576	−0.120288	+	0.992739i	$0.538382\pi$
$192$	0	0
$193$	18.3726	1.32249	0.661243	−	0.750172i	$-0.270029\pi$
$193$	18.3726	1.32249	0.661243	+	0.750172i	$0.270029\pi$
$194$	32.3470	2.32238
$195$	0	0
$196$	0	0
$197$	8.11567	0.578218	0.289109	−	0.957296i	$-0.406641\pi$
$197$	8.11567	0.578218	0.289109	+	0.957296i	$0.406641\pi$
$198$	0	0
$199$	6.52387	0.462465	0.231232	−	0.972899i	$-0.425724\pi$
$199$	6.52387	0.462465	0.231232	+	0.972899i	$0.425724\pi$
$200$	30.1812	2.13414
$201$	0	0
$202$	24.2569	1.70671
$203$	0	0
$204$	0	0
$205$	1.58170	0.110471
$206$	24.9065	1.73532
$207$	0	0
$208$	−0.871558	−0.0604317
$209$	−0.924344	−0.0639382
$210$	0	0
$211$	13.8487	0.953383	0.476691	−	0.879071i	$-0.341836\pi$
$211$	13.8487	0.953383	0.476691	+	0.879071i	$0.341836\pi$
$212$	21.0578	1.44626
$213$	0	0
$214$	2.33759	0.159794
$215$	−0.408196	−0.0278388
$216$	0	0
$217$	0	0
$218$	21.5562	1.45997
$219$	0	0
$220$	−0.586754	−0.0395589
$221$	0.675180	0.0454175
$222$	0	0
$223$	24.4983	1.64053	0.820265	−	0.571984i	$-0.193826\pi$
$223$	24.4983	1.64053	0.820265	+	0.571984i	$0.193826\pi$
$224$	0	0
$225$	0	0
$226$	−30.6395	−2.03811
$227$	−7.73302	−0.513258	−0.256629	−	0.966510i	$-0.582612\pi$
$227$	−7.73302	−0.513258	−0.256629	+	0.966510i	$0.582612\pi$
$228$	0	0
$229$	4.79085	0.316588	0.158294	−	0.987392i	$-0.449401\pi$
$229$	4.79085	0.316588	0.158294	+	0.987392i	$0.449401\pi$
$230$	−2.38266	−0.157108
$231$	0	0
$232$	23.4227	1.53777
$233$	9.69738	0.635296	0.317648	−	0.948209i	$-0.397107\pi$
$233$	9.69738	0.635296	0.317648	+	0.948209i	$0.397107\pi$
$234$	0	0
$235$	−0.410575	−0.0267830
$236$	−55.6345	−3.62150
$237$	0	0
$238$	0	0
$239$	−23.3070	−1.50760	−0.753802	−	0.657101i	$-0.771782\pi$
$239$	−23.3070	−1.50760	−0.753802	+	0.657101i	$0.771782\pi$
$240$	0	0
$241$	−9.71520	−0.625811	−0.312905	−	0.949784i	$-0.601302\pi$
$241$	−9.71520	−0.625811	−0.312905	+	0.949784i	$0.601302\pi$
$242$	−2.52892	−0.162565
$243$	0	0
$244$	−26.3726	−1.68833
$245$	0	0
$246$	0	0
$247$	−0.123392	−0.00785127
$248$	16.9065	1.07357
$249$	0	0
$250$	3.36989	0.213130
$251$	−15.0400	−0.949317	−0.474659	−	0.880170i	$-0.657428\pi$
$251$	−15.0400	−0.949317	−0.474659	+	0.880170i	$0.657428\pi$
$252$	0	0
$253$	−7.05784	−0.443722
$254$	22.5239	1.41327
$255$	0	0
$256$	−30.7680	−1.92300
$257$	15.8309	0.987502	0.493751	−	0.869603i	$-0.335625\pi$
$257$	15.8309	0.987502	0.493751	+	0.869603i	$0.335625\pi$
$258$	0	0
$259$	0	0
$260$	−0.0783269	−0.00485763
$261$	0	0
$262$	39.6974	2.45251
$263$	3.07566	0.189653	0.0948265	−	0.995494i	$-0.469770\pi$
$263$	3.07566	0.189653	0.0948265	+	0.995494i	$0.469770\pi$
$264$	0	0
$265$	0.639540	0.0392866
$266$	0	0
$267$	0	0
$268$	39.2263	2.39613
$269$	−10.5340	−0.642267	−0.321134	−	0.947034i	$-0.604064\pi$
$269$	−10.5340	−0.642267	−0.321134	+	0.947034i	$0.604064\pi$
$270$	0	0
$271$	4.92434	0.299133	0.149566	−	0.988752i	$-0.452212\pi$
$271$	4.92434	0.299133	0.149566	+	0.988752i	$0.452212\pi$
$272$	−33.0222	−2.00226
$273$	0	0
$274$	37.0222	2.23659
$275$	4.98218	0.300437
$276$	0	0
$277$	−0.151312	−0.00909146	−0.00454573	−	0.999990i	$-0.501447\pi$
$277$	−0.151312	−0.00909146	−0.00454573	+	0.999990i	$0.501447\pi$
$278$	47.1634	2.82867
$279$	0	0
$280$	0	0
$281$	−6.51615	−0.388721	−0.194360	−	0.980930i	$-0.562263\pi$
$281$	−6.51615	−0.388721	−0.194360	+	0.980930i	$0.562263\pi$
$282$	0	0
$283$	0.390376	0.0232055	0.0116027	−	0.999933i	$-0.496307\pi$
$283$	0.390376	0.0232055	0.0116027	+	0.999933i	$0.496307\pi$
$284$	26.8810	1.59509
$285$	0	0
$286$	−0.337590	−0.0199621
$287$	0	0
$288$	0	0
$289$	8.58170	0.504806
$290$	1.30529	0.0766496
$291$	0	0
$292$	−34.5767	−2.02345
$293$	29.4304	1.71934	0.859671	−	0.510848i	$-0.170669\pi$
$293$	29.4304	1.71934	0.859671	+	0.510848i	$0.170669\pi$
$294$	0	0
$295$	−1.68966	−0.0983755
$296$	−60.4704	−3.51477
$297$	0	0
$298$	30.0094	1.73840
$299$	−0.942164	−0.0544868
$300$	0	0
$301$	0	0
$302$	28.6395	1.64802
$303$	0	0
$304$	6.03497	0.346129
$305$	−0.800952	−0.0458624
$306$	0	0
$307$	−23.6974	−1.35248	−0.676240	−	0.736681i	$-0.736392\pi$
$307$	−23.6974	−1.35248	−0.676240	+	0.736681i	$0.736392\pi$
$308$	0	0
$309$	0	0
$310$	0.942164	0.0535113
$311$	−12.2313	−0.693576	−0.346788	−	0.937944i	$-0.612728\pi$
$311$	−12.2313	−0.693576	−0.346788	+	0.937944i	$0.612728\pi$
$312$	0	0
$313$	19.1735	1.08375	0.541875	−	0.840459i	$-0.317714\pi$
$313$	19.1735	1.08375	0.541875	+	0.840459i	$0.317714\pi$
$314$	−51.5461	−2.90891
$315$	0	0
$316$	62.0444	3.49027
$317$	−29.5562	−1.66004	−0.830020	−	0.557734i	$-0.811671\pi$
$317$	−29.5562	−1.66004	−0.830020	+	0.557734i	$0.811671\pi$
$318$	0	0
$319$	3.86651	0.216483
$320$	−0.259263	−0.0144933
$321$	0	0
$322$	0	0
$323$	−4.67518	−0.260134
$324$	0	0
$325$	0.665081	0.0368920
$326$	48.8258	2.70421
$327$	0	0
$328$	−71.7774	−3.96324
$329$	0	0
$330$	0	0
$331$	18.6496	1.02508	0.512538	−	0.858664i	$-0.328705\pi$
$331$	18.6496	1.02508	0.512538	+	0.858664i	$0.328705\pi$
$332$	−5.31472	−0.291683
$333$	0	0
$334$	−31.9644	−1.74901
$335$	1.19133	0.0650892
$336$	0	0
$337$	−21.0222	−1.14515	−0.572576	−	0.819852i	$-0.694056\pi$
$337$	−21.0222	−1.14515	−0.572576	+	0.819852i	$0.694056\pi$
$338$	32.8309	1.78576
$339$	0	0
$340$	−2.96770	−0.160946
$341$	2.79085	0.151133
$342$	0	0
$343$	0	0
$344$	18.5239	0.998740
$345$	0	0
$346$	50.1957	2.69854
$347$	−23.1634	−1.24348	−0.621738	−	0.783225i	$-0.713573\pi$
$347$	−23.1634	−1.24348	−0.621738	+	0.783225i	$0.713573\pi$
$348$	0	0
$349$	28.0979	1.50404	0.752022	−	0.659138i	$-0.229079\pi$
$349$	28.0979	1.50404	0.752022	+	0.659138i	$0.229079\pi$
$350$	0	0
$351$	0	0
$352$	4.39543	0.234277
$353$	−33.4126	−1.77837	−0.889186	−	0.457546i	$-0.848728\pi$
$353$	−33.4126	−1.77837	−0.889186	+	0.457546i	$0.848728\pi$
$354$	0	0
$355$	0.816393	0.0433296
$356$	68.4882	3.62987
$357$	0	0
$358$	−30.3470	−1.60389
$359$	−13.5817	−0.716815	−0.358407	−	0.933565i	$-0.616680\pi$
$359$	−13.5817	−0.716815	−0.358407	+	0.933565i	$0.616680\pi$
$360$	0	0
$361$	−18.1456	−0.955031
$362$	6.76531	0.355577
$363$	0	0
$364$	0	0
$365$	−1.05012	−0.0549655
$366$	0	0
$367$	3.73302	0.194862	0.0974309	−	0.995242i	$-0.468937\pi$
$367$	3.73302	0.194862	0.0974309	+	0.995242i	$0.468937\pi$
$368$	46.0800	2.40209
$369$	0	0
$370$	−3.36989	−0.175192
$371$	0	0
$372$	0	0
$373$	6.94216	0.359452	0.179726	−	0.983717i	$-0.442479\pi$
$373$	6.94216	0.359452	0.179726	+	0.983717i	$0.442479\pi$
$374$	−12.7909	−0.661399
$375$	0	0
$376$	18.6318	0.960863
$377$	0.516148	0.0265830
$378$	0	0
$379$	−0.390376	−0.0200523	−0.0100261	−	0.999950i	$-0.503191\pi$
$379$	−0.390376	−0.0200523	−0.0100261	+	0.999950i	$0.503191\pi$
$380$	0.542362	0.0278226
$381$	0	0
$382$	8.40820	0.430201
$383$	−0.533968	−0.0272845	−0.0136422	−	0.999907i	$-0.504343\pi$
$383$	−0.533968	−0.0272845	−0.0136422	+	0.999907i	$0.504343\pi$
$384$	0	0
$385$	0	0
$386$	−46.4627	−2.36489
$387$	0	0
$388$	−56.2212	−2.85420
$389$	10.4983	0.532286	0.266143	−	0.963934i	$-0.414251\pi$
$389$	10.4983	0.532286	0.266143	+	0.963934i	$0.414251\pi$
$390$	0	0
$391$	−35.6974	−1.80529
$392$	0	0
$393$	0	0
$394$	−20.5239	−1.03398
$395$	1.88433	0.0948108
$396$	0	0
$397$	20.7552	1.04167	0.520837	−	0.853656i	$-0.325620\pi$
$397$	20.7552	1.04167	0.520837	+	0.853656i	$0.325620\pi$
$398$	−16.4983	−0.826986
$399$	0	0
$400$	−32.5282	−1.62641
$401$	−21.3248	−1.06491	−0.532455	−	0.846458i	$-0.678731\pi$
$401$	−21.3248	−1.06491	−0.532455	+	0.846458i	$0.678731\pi$
$402$	0	0
$403$	0.372556	0.0185583
$404$	−42.1601	−2.09754
$405$	0	0
$406$	0	0
$407$	−9.98218	−0.494798
$408$	0	0
$409$	−33.9287	−1.67767	−0.838834	−	0.544388i	$-0.816762\pi$
$409$	−33.9287	−1.67767	−0.838834	+	0.544388i	$0.816762\pi$
$410$	−4.00000	−0.197546
$411$	0	0
$412$	−43.2892	−2.13270
$413$	0	0
$414$	0	0
$415$	−0.161411	−0.00792338
$416$	0.586754	0.0287680
$417$	0	0
$418$	2.33759	0.114335
$419$	−9.99228	−0.488155	−0.244077	−	0.969756i	$-0.578485\pi$
$419$	−9.99228	−0.488155	−0.244077	+	0.969756i	$0.578485\pi$
$420$	0	0
$421$	−29.9822	−1.46124	−0.730621	−	0.682783i	$-0.760769\pi$
$421$	−29.9822	−1.46124	−0.730621	+	0.682783i	$0.760769\pi$
$422$	−35.0222	−1.70485
$423$	0	0
$424$	−29.0222	−1.40944
$425$	25.1990	1.22233
$426$	0	0
$427$	0	0
$428$	−4.06289	−0.196387
$429$	0	0
$430$	1.03230	0.0497817
$431$	2.54169	0.122429	0.0612144	−	0.998125i	$-0.480503\pi$
$431$	2.54169	0.122429	0.0612144	+	0.998125i	$0.480503\pi$
$432$	0	0
$433$	−20.3827	−0.979528	−0.489764	−	0.871855i	$-0.662917\pi$
$433$	−20.3827	−0.979528	−0.489764	+	0.871855i	$0.662917\pi$
$434$	0	0
$435$	0	0
$436$	−37.4660	−1.79430
$437$	6.52387	0.312079
$438$	0	0
$439$	5.45831	0.260511	0.130256	−	0.991480i	$-0.458420\pi$
$439$	5.45831	0.260511	0.130256	+	0.991480i	$0.458420\pi$
$440$	0.808672	0.0385519
$441$	0	0
$442$	−1.70748	−0.0812163
$443$	10.6496	0.505980	0.252990	−	0.967469i	$-0.418586\pi$
$443$	10.6496	0.505980	0.252990	+	0.967469i	$0.418586\pi$
$444$	0	0
$445$	2.08003	0.0986030
$446$	−61.9543	−2.93362
$447$	0	0
$448$	0	0
$449$	−27.0323	−1.27573	−0.637866	−	0.770147i	$-0.720183\pi$
$449$	−27.0323	−1.27573	−0.637866	+	0.770147i	$0.720183\pi$
$450$	0	0
$451$	−11.8487	−0.557933
$452$	53.2535	2.50484
$453$	0	0
$454$	19.5562	0.917816
$455$	0	0
$456$	0	0
$457$	4.79085	0.224107	0.112053	−	0.993702i	$-0.464257\pi$
$457$	4.79085	0.224107	0.112053	+	0.993702i	$0.464257\pi$
$458$	−12.1157	−0.566128
$459$	0	0
$460$	4.14121	0.193085
$461$	22.4983	1.04785	0.523926	−	0.851764i	$-0.324467\pi$
$461$	22.4983	1.04785	0.523926	+	0.851764i	$0.324467\pi$
$462$	0	0
$463$	25.6897	1.19390	0.596950	−	0.802279i	$-0.296379\pi$
$463$	25.6897	1.19390	0.596950	+	0.802279i	$0.296379\pi$
$464$	−25.2441	−1.17193
$465$	0	0
$466$	−24.5239	−1.13605
$467$	4.62172	0.213868	0.106934	−	0.994266i	$-0.465897\pi$
$467$	4.62172	0.213868	0.106934	+	0.994266i	$0.465897\pi$
$468$	0	0
$469$	0	0
$470$	1.03831	0.0478937
$471$	0	0
$472$	76.6762	3.52931
$473$	3.05784	0.140599
$474$	0	0
$475$	−4.60525	−0.211303
$476$	0	0
$477$	0	0
$478$	58.9415	2.69592
$479$	−0.372556	−0.0170225	−0.00851126	−	0.999964i	$-0.502709\pi$
$479$	−0.372556	−0.0170225	−0.00851126	+	0.999964i	$0.502709\pi$
$480$	0	0
$481$	−1.33254	−0.0607586
$482$	24.5689	1.11908
$483$	0	0
$484$	4.39543	0.199792
$485$	−1.70748	−0.0775325
$486$	0	0
$487$	4.30262	0.194971	0.0974853	−	0.995237i	$-0.468920\pi$
$487$	4.30262	0.194971	0.0974853	+	0.995237i	$0.468920\pi$
$488$	36.3470	1.64535
$489$	0	0
$490$	0	0
$491$	−25.4583	−1.14892	−0.574459	−	0.818534i	$-0.694787\pi$
$491$	−25.4583	−1.14892	−0.574459	+	0.818534i	$0.694787\pi$
$492$	0	0
$493$	19.5562	0.880765
$494$	0.312049	0.0140398
$495$	0	0
$496$	−18.2212	−0.818158
$497$	0	0
$498$	0	0
$499$	−8.39038	−0.375605	−0.187802	−	0.982207i	$-0.560136\pi$
$499$	−8.39038	−0.375605	−0.187802	+	0.982207i	$0.560136\pi$
$500$	−5.85708	−0.261937
$501$	0	0
$502$	38.0350	1.69758
$503$	−26.7552	−1.19296	−0.596478	−	0.802629i	$-0.703434\pi$
$503$	−26.7552	−1.19296	−0.596478	+	0.802629i	$0.703434\pi$
$504$	0	0
$505$	−1.28043	−0.0569783
$506$	17.8487	0.793471
$507$	0	0
$508$	−39.1480	−1.73691
$509$	−18.0000	−0.797836	−0.398918	−	0.916987i	$-0.630614\pi$
$509$	−18.0000	−0.797836	−0.398918	+	0.916987i	$0.630614\pi$
$510$	0	0
$511$	0	0
$512$	50.4049	2.22760
$513$	0	0
$514$	−40.0350	−1.76587
$515$	−1.31472	−0.0579335
$516$	0	0
$517$	3.07566	0.135267
$518$	0	0
$519$	0	0
$520$	0.107951	0.00473397
$521$	−1.44821	−0.0634473	−0.0317237	−	0.999497i	$-0.510100\pi$
$521$	−1.44821	−0.0634473	−0.0317237	+	0.999497i	$0.510100\pi$
$522$	0	0
$523$	−23.5740	−1.03082	−0.515409	−	0.856944i	$-0.672360\pi$
$523$	−23.5740	−1.03082	−0.515409	+	0.856944i	$0.672360\pi$
$524$	−68.9967	−3.01413
$525$	0	0
$526$	−7.77808	−0.339140
$527$	14.1157	0.614888
$528$	0	0
$529$	26.8130	1.16578
$530$	−1.61734	−0.0702529
$531$	0	0
$532$	0	0
$533$	−1.58170	−0.0685112
$534$	0	0
$535$	−0.123392	−0.00533472
$536$	−54.0622	−2.33513
$537$	0	0
$538$	26.6395	1.14851
$539$	0	0
$540$	0	0
$541$	−18.4983	−0.795305	−0.397653	−	0.917536i	$-0.630175\pi$
$541$	−18.4983	−0.795305	−0.397653	+	0.917536i	$0.630175\pi$
$542$	−12.4533	−0.534913
$543$	0	0
$544$	22.2313	0.953161
$545$	−1.13787	−0.0487409
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	−64.3470	−2.74877
$549$	0	0
$550$	−12.5995	−0.537246
$551$	−3.57398	−0.152257
$552$	0	0
$553$	0	0
$554$	0.382656	0.0162575
$555$	0	0
$556$	−81.9731	−3.47643
$557$	−1.98218	−0.0839877	−0.0419938	−	0.999118i	$-0.513371\pi$
$557$	−1.98218	−0.0839877	−0.0419938	+	0.999118i	$0.513371\pi$
$558$	0	0
$559$	0.408196	0.0172649
$560$	0	0
$561$	0	0
$562$	16.4788	0.695116
$563$	1.98990	0.0838643	0.0419322	−	0.999120i	$-0.486649\pi$
$563$	1.98990	0.0838643	0.0419322	+	0.999120i	$0.486649\pi$
$564$	0	0
$565$	1.61734	0.0680422
$566$	−0.987230	−0.0414964
$567$	0	0
$568$	−37.0477	−1.55449
$569$	14.5340	0.609296	0.304648	−	0.952465i	$-0.401461\pi$
$569$	14.5340	0.609296	0.304648	+	0.952465i	$0.401461\pi$
$570$	0	0
$571$	−29.2791	−1.22529	−0.612646	−	0.790358i	$-0.709895\pi$
$571$	−29.2791	−1.22529	−0.612646	+	0.790358i	$0.709895\pi$
$572$	0.586754	0.0245334
$573$	0	0
$574$	0	0
$575$	−35.1634	−1.46642
$576$	0	0
$577$	5.59180	0.232790	0.116395	−	0.993203i	$-0.462866\pi$
$577$	5.59180	0.232790	0.116395	+	0.993203i	$0.462866\pi$
$578$	−21.7024	−0.902702
$579$	0	0
$580$	−2.26869	−0.0942022
$581$	0	0
$582$	0	0
$583$	−4.79085	−0.198417
$584$	47.6540	1.97194
$585$	0	0
$586$	−74.4270	−3.07455
$587$	−2.80867	−0.115926	−0.0579632	−	0.998319i	$-0.518461\pi$
$587$	−2.80867	−0.115926	−0.0579632	+	0.998319i	$0.518461\pi$
$588$	0	0
$589$	−2.57971	−0.106295
$590$	4.27300	0.175917
$591$	0	0
$592$	65.1728	2.67859
$593$	−42.1957	−1.73277	−0.866385	−	0.499377i	$-0.833562\pi$
$593$	−42.1957	−1.73277	−0.866385	+	0.499377i	$0.833562\pi$
$594$	0	0
$595$	0	0
$596$	−52.1584	−2.13649
$597$	0	0
$598$	2.38266	0.0974340
$599$	−34.3470	−1.40338	−0.701691	−	0.712482i	$-0.747571\pi$
$599$	−34.3470	−1.40338	−0.701691	+	0.712482i	$0.747571\pi$
$600$	0	0
$601$	−5.98218	−0.244018	−0.122009	−	0.992529i	$-0.538934\pi$
$601$	−5.98218	−0.244018	−0.122009	+	0.992529i	$0.538934\pi$
$602$	0	0
$603$	0	0
$604$	−49.7774	−2.02541
$605$	0.133492	0.00542722
$606$	0	0
$607$	−8.12339	−0.329718	−0.164859	−	0.986317i	$-0.552717\pi$
$607$	−8.12339	−0.329718	−0.164859	+	0.986317i	$0.552717\pi$
$608$	−4.06289	−0.164772
$609$	0	0
$610$	2.02554	0.0820117
$611$	0.410575	0.0166101
$612$	0	0
$613$	−16.5239	−0.667393	−0.333696	−	0.942681i	$-0.608296\pi$
$613$	−16.5239	−0.667393	−0.333696	+	0.942681i	$0.608296\pi$
$614$	59.9287	2.41853
$615$	0	0
$616$	0	0
$617$	−11.8843	−0.478445	−0.239223	−	0.970965i	$-0.576893\pi$
$617$	−11.8843	−0.478445	−0.239223	+	0.970965i	$0.576893\pi$
$618$	0	0
$619$	−43.0222	−1.72921	−0.864604	−	0.502454i	$-0.832431\pi$
$619$	−43.0222	−1.72921	−0.864604	+	0.502454i	$0.832431\pi$
$620$	−1.63754	−0.0657653
$621$	0	0
$622$	30.9321	1.24026
$623$	0	0
$624$	0	0
$625$	24.7330	0.989321
$626$	−48.4882	−1.93798
$627$	0	0
$628$	89.5905	3.57505
$629$	−50.4882	−2.01310
$630$	0	0
$631$	13.5817	0.540679	0.270340	−	0.962765i	$-0.412864\pi$
$631$	13.5817	0.540679	0.270340	+	0.962765i	$0.412864\pi$
$632$	−85.5104	−3.40142
$633$	0	0
$634$	74.7451	2.96851
$635$	−1.18895	−0.0471820
$636$	0	0
$637$	0	0
$638$	−9.77808	−0.387118
$639$	0	0
$640$	1.82916	0.0723040
$641$	−36.7552	−1.45174	−0.725872	−	0.687830i	$-0.758563\pi$
$641$	−36.7552	−1.45174	−0.725872	+	0.687830i	$0.758563\pi$
$642$	0	0
$643$	44.3369	1.74848	0.874239	−	0.485496i	$-0.161361\pi$
$643$	44.3369	1.74848	0.874239	+	0.485496i	$0.161361\pi$
$644$	0	0
$645$	0	0
$646$	11.8231	0.465176
$647$	−17.9721	−0.706555	−0.353278	−	0.935519i	$-0.614933\pi$
$647$	−17.9721	−0.706555	−0.353278	+	0.935519i	$0.614933\pi$
$648$	0	0
$649$	12.6574	0.496845
$650$	−1.68193	−0.0659709
$651$	0	0
$652$	−84.8625	−3.32347
$653$	22.1957	0.868585	0.434292	−	0.900772i	$-0.356998\pi$
$653$	22.1957	0.868585	0.434292	+	0.900772i	$0.356998\pi$
$654$	0	0
$655$	−2.09547	−0.0818769
$656$	77.3591	3.02037
$657$	0	0
$658$	0	0
$659$	−9.99228	−0.389244	−0.194622	−	0.980878i	$-0.562348\pi$
$659$	−9.99228	−0.389244	−0.194622	+	0.980878i	$0.562348\pi$
$660$	0	0
$661$	−27.9543	−1.08729	−0.543647	−	0.839314i	$-0.682957\pi$
$661$	−27.9543	−1.08729	−0.543647	+	0.839314i	$0.682957\pi$
$662$	−47.1634	−1.83306
$663$	0	0
$664$	7.32482	0.284258
$665$	0	0
$666$	0	0
$667$	−27.2892	−1.05664
$668$	55.5562	2.14953
$669$	0	0
$670$	−3.01277	−0.116393
$671$	6.00000	0.231627
$672$	0	0
$673$	−43.4203	−1.67373	−0.836865	−	0.547410i	$-0.815614\pi$
$673$	−43.4203	−1.67373	−0.836865	+	0.547410i	$0.815614\pi$
$674$	53.1634	2.04778
$675$	0	0
$676$	−57.0622	−2.19470
$677$	−12.3470	−0.474534	−0.237267	−	0.971444i	$-0.576252\pi$
$677$	−12.3470	−0.474534	−0.237267	+	0.971444i	$0.576252\pi$
$678$	0	0
$679$	0	0
$680$	4.09013	0.156849
$681$	0	0
$682$	−7.05784	−0.270259
$683$	27.8386	1.06521	0.532607	−	0.846363i	$-0.321212\pi$
$683$	27.8386	1.06521	0.532607	+	0.846363i	$0.321212\pi$
$684$	0	0
$685$	−1.95426	−0.0746684
$686$	0	0
$687$	0	0
$688$	−19.9644	−0.761134
$689$	−0.639540	−0.0243645
$690$	0	0
$691$	16.8010	0.639138	0.319569	−	0.947563i	$-0.396462\pi$
$691$	16.8010	0.639138	0.319569	+	0.947563i	$0.396462\pi$
$692$	−87.2434	−3.31650
$693$	0	0
$694$	58.5784	2.22360
$695$	−2.48958	−0.0944350
$696$	0	0
$697$	−59.9287	−2.26996
$698$	−71.0572	−2.68955
$699$	0	0
$700$	0	0
$701$	−9.16341	−0.346097	−0.173049	−	0.984913i	$-0.555362\pi$
$701$	−9.16341	−0.346097	−0.173049	+	0.984913i	$0.555362\pi$
$702$	0	0
$703$	9.22697	0.348002
$704$	1.94216	0.0731981
$705$	0	0
$706$	84.4977	3.18011
$707$	0	0
$708$	0	0
$709$	−34.7475	−1.30497	−0.652485	−	0.757802i	$-0.726273\pi$
$709$	−34.7475	−1.30497	−0.652485	+	0.757802i	$0.726273\pi$
$710$	−2.06459	−0.0774827
$711$	0	0
$712$	−94.3914	−3.53747
$713$	−19.6974	−0.737673
$714$	0	0
$715$	0.0178201	0.000666434 0
$716$	52.7451	1.97118
$717$	0	0
$718$	34.3470	1.28182
$719$	−25.1557	−0.938149	−0.469074	−	0.883159i	$-0.655412\pi$
$719$	−25.1557	−0.938149	−0.469074	+	0.883159i	$0.655412\pi$
$720$	0	0
$721$	0	0
$722$	45.8887	1.70780
$723$	0	0
$724$	−11.7586	−0.437003
$725$	19.2636	0.715434
$726$	0	0
$727$	23.5918	0.874972	0.437486	−	0.899225i	$-0.355869\pi$
$727$	23.5918	0.874972	0.437486	+	0.899225i	$0.355869\pi$
$728$	0	0
$729$	0	0
$730$	2.65566	0.0982902
$731$	15.4660	0.572032
$732$	0	0
$733$	−21.9287	−0.809956	−0.404978	−	0.914326i	$-0.632721\pi$
$733$	−21.9287	−0.809956	−0.404978	+	0.914326i	$0.632721\pi$
$734$	−9.44049	−0.348455
$735$	0	0
$736$	−31.0222	−1.14349
$737$	−8.92434	−0.328732
$738$	0	0
$739$	35.0578	1.28962	0.644812	−	0.764341i	$-0.276936\pi$
$739$	35.0578	1.28962	0.644812	+	0.764341i	$0.276936\pi$
$740$	5.85708	0.215311
$741$	0	0
$742$	0	0
$743$	31.3070	1.14854	0.574271	−	0.818665i	$-0.305285\pi$
$743$	31.3070	1.14854	0.574271	+	0.818665i	$0.305285\pi$
$744$	0	0
$745$	−1.58408	−0.0580363
$746$	−17.5562	−0.642777
$747$	0	0
$748$	22.2313	0.812858
$749$	0	0
$750$	0	0
$751$	−4.42602	−0.161508	−0.0807538	−	0.996734i	$-0.525733\pi$
$751$	−4.42602	−0.161508	−0.0807538	+	0.996734i	$0.525733\pi$
$752$	−20.0807	−0.732268
$753$	0	0
$754$	−1.30529	−0.0475360
$755$	−1.51177	−0.0550190
$756$	0	0
$757$	2.51615	0.0914509	0.0457255	−	0.998954i	$-0.485440\pi$
$757$	2.51615	0.0914509	0.0457255	+	0.998954i	$0.485440\pi$
$758$	0.987230	0.0358578
$759$	0	0
$760$	−0.747491	−0.0271144
$761$	13.7330	0.497821	0.248911	−	0.968526i	$-0.419927\pi$
$761$	13.7330	0.497821	0.248911	+	0.968526i	$0.419927\pi$
$762$	0	0
$763$	0	0
$764$	−14.6140	−0.528716
$765$	0	0
$766$	1.35036	0.0487905
$767$	1.68966	0.0610099
$768$	0	0
$769$	−46.4805	−1.67613	−0.838065	−	0.545570i	$-0.816313\pi$
$769$	−46.4805	−1.67613	−0.838065	+	0.545570i	$0.816313\pi$
$770$	0	0
$771$	0	0
$772$	80.7552	2.90644
$773$	−45.9822	−1.65386	−0.826932	−	0.562302i	$-0.809916\pi$
$773$	−45.9822	−1.65386	−0.826932	+	0.562302i	$0.809916\pi$
$774$	0	0
$775$	13.9045	0.499465
$776$	77.4849	2.78155
$777$	0	0
$778$	−26.5494	−0.951842
$779$	10.9523	0.392406
$780$	0	0
$781$	−6.11567	−0.218836
$782$	90.2757	3.22825
$783$	0	0
$784$	0	0
$785$	2.72092	0.0971138
$786$	0	0
$787$	−40.9243	−1.45880	−0.729398	−	0.684090i	$-0.760200\pi$
$787$	−40.9243	−1.45880	−0.729398	+	0.684090i	$0.760200\pi$
$788$	35.6718	1.27076
$789$	0	0
$790$	−4.76531	−0.169542
$791$	0	0
$792$	0	0
$793$	0.800952	0.0284426
$794$	−52.4882	−1.86274
$795$	0	0
$796$	28.6752	1.01636
$797$	−28.8786	−1.02293	−0.511466	−	0.859303i	$-0.670898\pi$
$797$	−28.8786	−1.02293	−0.511466	+	0.859303i	$0.670898\pi$
$798$	0	0
$799$	15.5562	0.550338
$800$	21.8988	0.774240
$801$	0	0
$802$	53.9287	1.90429
$803$	7.86651	0.277603
$804$	0	0
$805$	0	0
$806$	−0.942164	−0.0331863
$807$	0	0
$808$	58.1056	2.04415
$809$	−6.51615	−0.229096	−0.114548	−	0.993418i	$-0.536542\pi$
$809$	−6.51615	−0.229096	−0.114548	+	0.993418i	$0.536542\pi$
$810$	0	0
$811$	−38.5060	−1.35213	−0.676065	−	0.736842i	$-0.736316\pi$
$811$	−38.5060	−1.35213	−0.676065	+	0.736842i	$0.736316\pi$
$812$	0	0
$813$	0	0
$814$	25.2441	0.884806
$815$	−2.57733	−0.0902799
$816$	0	0
$817$	−2.82649	−0.0988864
$818$	85.8029	3.00003
$819$	0	0
$820$	6.95226	0.242784
$821$	37.3769	1.30446	0.652232	−	0.758019i	$-0.273833\pi$
$821$	37.3769	1.30446	0.652232	+	0.758019i	$0.273833\pi$
$822$	0	0
$823$	43.0400	1.50028	0.750140	−	0.661279i	$-0.229986\pi$
$823$	43.0400	1.50028	0.750140	+	0.661279i	$0.229986\pi$
$824$	59.6617	2.07842
$825$	0	0
$826$	0	0
$827$	−11.3426	−0.394422	−0.197211	−	0.980361i	$-0.563188\pi$
$827$	−11.3426	−0.394422	−0.197211	+	0.980361i	$0.563188\pi$
$828$	0	0
$829$	−14.4983	−0.503548	−0.251774	−	0.967786i	$-0.581014\pi$
$829$	−14.4983	−0.503548	−0.251774	+	0.967786i	$0.581014\pi$
$830$	0.408196	0.0141687
$831$	0	0
$832$	0.259263	0.00898833
$833$	0	0
$834$	0	0
$835$	1.68728	0.0583906
$836$	−4.06289	−0.140518
$837$	0	0
$838$	25.2697	0.872926
$839$	−3.60962	−0.124618	−0.0623090	−	0.998057i	$-0.519846\pi$
$839$	−3.60962	−0.124618	−0.0623090	+	0.998057i	$0.519846\pi$
$840$	0	0
$841$	−14.0501	−0.484487
$842$	75.8225	2.61301
$843$	0	0
$844$	60.8709	2.09526
$845$	−1.73302	−0.0596176
$846$	0	0
$847$	0	0
$848$	31.2791	1.07413
$849$	0	0
$850$	−63.7263	−2.18579
$851$	70.4526	2.41508
$852$	0	0
$853$	24.6496	0.843988	0.421994	−	0.906599i	$-0.361330\pi$
$853$	24.6496	0.843988	0.421994	+	0.906599i	$0.361330\pi$
$854$	0	0
$855$	0	0
$856$	5.59952	0.191388
$857$	17.9287	0.612433	0.306217	−	0.951962i	$-0.400937\pi$
$857$	17.9287	0.612433	0.306217	+	0.951962i	$0.400937\pi$
$858$	0	0
$859$	15.6974	0.535588	0.267794	−	0.963476i	$-0.413705\pi$
$859$	15.6974	0.535588	0.267794	+	0.963476i	$0.413705\pi$
$860$	−1.79420	−0.0611816
$861$	0	0
$862$	−6.42772	−0.218929
$863$	15.0578	0.512575	0.256287	−	0.966601i	$-0.417501\pi$
$863$	15.0578	0.512575	0.256287	+	0.966601i	$0.417501\pi$
$864$	0	0
$865$	−2.64964	−0.0900904
$866$	51.5461	1.75161
$867$	0	0
$868$	0	0
$869$	−14.1157	−0.478841
$870$	0	0
$871$	−1.19133	−0.0403666
$872$	51.6362	1.74862
$873$	0	0
$874$	−16.4983	−0.558064
$875$	0	0
$876$	0	0
$877$	−43.8487	−1.48066	−0.740332	−	0.672241i	$-0.765332\pi$
$877$	−43.8487	−1.48066	−0.740332	+	0.672241i	$0.765332\pi$
$878$	−13.8036	−0.465850
$879$	0	0
$880$	−0.871558	−0.0293802
$881$	11.0656	0.372808	0.186404	−	0.982473i	$-0.440317\pi$
$881$	11.0656	0.372808	0.186404	+	0.982473i	$0.440317\pi$
$882$	0	0
$883$	5.72530	0.192672	0.0963358	−	0.995349i	$-0.469288\pi$
$883$	5.72530	0.192672	0.0963358	+	0.995349i	$0.469288\pi$
$884$	2.96770	0.0998147
$885$	0	0
$886$	−26.9321	−0.904800
$887$	15.0935	0.506789	0.253395	−	0.967363i	$-0.418453\pi$
$887$	15.0935	0.506789	0.253395	+	0.967363i	$0.418453\pi$
$888$	0	0
$889$	0	0
$890$	−5.26023	−0.176323
$891$	0	0
$892$	107.681	3.60541
$893$	−2.84296	−0.0951362
$894$	0	0
$895$	1.60190	0.0535457
$896$	0	0
$897$	0	0
$898$	68.3625	2.28128
$899$	10.7909	0.359895
$900$	0	0
$901$	−24.2313	−0.807263
$902$	29.9644	0.997704
$903$	0	0
$904$	−73.3948	−2.44107
$905$	−0.357115	−0.0118709
$906$	0	0
$907$	41.2993	1.37132	0.685660	−	0.727922i	$-0.259514\pi$
$907$	41.2993	1.37132	0.685660	+	0.727922i	$0.259514\pi$
$908$	−33.9899	−1.12799
$909$	0	0
$910$	0	0
$911$	−3.45059	−0.114323	−0.0571616	−	0.998365i	$-0.518205\pi$
$911$	−3.45059	−0.114323	−0.0571616	+	0.998365i	$0.518205\pi$
$912$	0	0
$913$	1.20915	0.0400170
$914$	−12.1157	−0.400751
$915$	0	0
$916$	21.0578	0.695770
$917$	0	0
$918$	0	0
$919$	−10.8265	−0.357133	−0.178567	−	0.983928i	$-0.557146\pi$
$919$	−10.8265	−0.357133	−0.178567	+	0.983928i	$0.557146\pi$
$920$	−5.70748	−0.188170
$921$	0	0
$922$	−56.8964	−1.87378
$923$	−0.816393	−0.0268719
$924$	0	0
$925$	−49.7330	−1.63521
$926$	−64.9670	−2.13495
$927$	0	0
$928$	16.9950	0.557887
$929$	17.4684	0.573120	0.286560	−	0.958062i	$-0.407488\pi$
$929$	17.4684	0.573120	0.286560	+	0.958062i	$0.407488\pi$
$930$	0	0
$931$	0	0
$932$	42.6241	1.39620
$933$	0	0
$934$	−11.6880	−0.382441
$935$	0.675180	0.0220808
$936$	0	0
$937$	−10.5340	−0.344130	−0.172065	−	0.985086i	$-0.555044\pi$
$937$	−10.5340	−0.344130	−0.172065	+	0.985086i	$0.555044\pi$
$938$	0	0
$939$	0	0
$940$	−1.80465	−0.0588613
$941$	−16.1513	−0.526518	−0.263259	−	0.964725i	$-0.584797\pi$
$941$	−16.1513	−0.526518	−0.263259	+	0.964725i	$0.584797\pi$
$942$	0	0
$943$	83.6261	2.72324
$944$	−82.6389	−2.68967
$945$	0	0
$946$	−7.73302	−0.251422
$947$	6.91662	0.224760	0.112380	−	0.993665i	$-0.464153\pi$
$947$	6.91662	0.224760	0.112380	+	0.993665i	$0.464153\pi$
$948$	0	0
$949$	1.05012	0.0340882
$950$	11.6463	0.377856
$951$	0	0
$952$	0	0
$953$	16.1335	0.522615	0.261308	−	0.965256i	$-0.415846\pi$
$953$	16.1335	0.522615	0.261308	+	0.965256i	$0.415846\pi$
$954$	0	0
$955$	−0.443837	−0.0143622
$956$	−102.444	−3.31328
$957$	0	0
$958$	0.942164	0.0304399
$959$	0	0
$960$	0	0
$961$	−23.2111	−0.748747
$962$	3.36989	0.108649
$963$	0	0
$964$	−42.7024	−1.37535
$965$	2.45259	0.0789516
$966$	0	0
$967$	17.8487	0.573975	0.286988	−	0.957934i	$-0.407346\pi$
$967$	17.8487	0.573975	0.286988	+	0.957934i	$0.407346\pi$
$968$	−6.05784	−0.194706
$969$	0	0
$970$	4.31807	0.138645
$971$	33.9566	1.08972	0.544860	−	0.838527i	$-0.316583\pi$
$971$	33.9566	1.08972	0.544860	+	0.838527i	$0.316583\pi$
$972$	0	0
$973$	0	0
$974$	−10.8810	−0.348649
$975$	0	0
$976$	−39.1735	−1.25391
$977$	−41.0222	−1.31242	−0.656208	−	0.754580i	$-0.727841\pi$
$977$	−41.0222	−1.31242	−0.656208	+	0.754580i	$0.727841\pi$
$978$	0	0
$979$	−15.5817	−0.497993
$980$	0	0
$981$	0	0
$982$	64.3820	2.05451
$983$	42.3470	1.35066	0.675330	−	0.737516i	$-0.264001\pi$
$983$	42.3470	1.35066	0.675330	+	0.737516i	$0.264001\pi$
$984$	0	0
$985$	1.08338	0.0345192
$986$	−49.4559	−1.57500
$987$	0	0
$988$	−0.542362	−0.0172548
$989$	−21.5817	−0.686258
$990$	0	0
$991$	50.9687	1.61908	0.809538	−	0.587068i	$-0.199718\pi$
$991$	50.9687	1.61908	0.809538	+	0.587068i	$0.199718\pi$
$992$	12.2670	0.389477
$993$	0	0
$994$	0	0
$995$	0.870884	0.0276089
$996$	0	0
$997$	32.8608	1.04071	0.520356	−	0.853950i	$-0.325799\pi$
$997$	32.8608	1.04071	0.520356	+	0.853950i	$0.325799\pi$
$998$	21.2186	0.671662
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4851.2.a.bp.1.1		3
3.2	odd	2		1617.2.a.s.1.3		3
7.6	odd	2		693.2.a.m.1.1		3
21.20	even	2		231.2.a.d.1.3	✓	3
77.76	even	2		7623.2.a.cb.1.3		3
84.83	odd	2		3696.2.a.bp.1.2		3
105.104	even	2		5775.2.a.bw.1.1		3
231.230	odd	2		2541.2.a.bi.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.a.d.1.3	✓	3	21.20	even	2
693.2.a.m.1.1		3	7.6	odd	2
1617.2.a.s.1.3		3	3.2	odd	2
2541.2.a.bi.1.1		3	231.230	odd	2
3696.2.a.bp.1.2		3	84.83	odd	2
4851.2.a.bp.1.1		3	1.1	even	1	trivial
5775.2.a.bw.1.1		3	105.104	even	2
7623.2.a.cb.1.3		3	77.76	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4851 = 3^{2} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4851.a (trivial)

\(f(q)\)	\(=\)	\(q-2.52892 q^{2} +4.39543 q^{4} +0.133492 q^{5} -6.05784 q^{8} +O(q^{10})\)	\(q-2.52892 q^{2} +4.39543 q^{4} +0.133492 q^{5} -6.05784 q^{8} -0.337590 q^{10} -1.00000 q^{11} -0.133492 q^{13} +6.52892 q^{16} -5.05784 q^{17} +0.924344 q^{19} +0.586754 q^{20} +2.52892 q^{22} +7.05784 q^{23} -4.98218 q^{25} +0.337590 q^{26} -3.86651 q^{29} -2.79085 q^{31} -4.39543 q^{32} +12.7909 q^{34} +9.98218 q^{37} -2.33759 q^{38} -0.808672 q^{40} +11.8487 q^{41} -3.05784 q^{43} -4.39543 q^{44} -17.8487 q^{46} -3.07566 q^{47} +12.5995 q^{50} -0.586754 q^{52} +4.79085 q^{53} -0.133492 q^{55} +9.77808 q^{58} -12.6574 q^{59} -6.00000 q^{61} +7.05784 q^{62} -1.94216 q^{64} -0.0178201 q^{65} +8.92434 q^{67} -22.2313 q^{68} +6.11567 q^{71} -7.86651 q^{73} -25.2441 q^{74} +4.06289 q^{76} +14.1157 q^{79} +0.871558 q^{80} -29.9644 q^{82} -1.20915 q^{83} -0.675180 q^{85} +7.73302 q^{86} +6.05784 q^{88} +15.5817 q^{89} +31.0222 q^{92} +7.77808 q^{94} +0.123392 q^{95} -12.7909 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{4} - 3 q^{8}+O(q^{10}) \) 3 * q + 6 * q^4 - 3 * q^8	\( 3 q + 6 q^{4} - 3 q^{8} - 9 q^{10} - 3 q^{11} + 12 q^{16} - 12 q^{19} - 21 q^{20} + 6 q^{23} + 15 q^{25} + 9 q^{26} - 12 q^{29} + 6 q^{31} - 6 q^{32} + 24 q^{34} - 15 q^{38} - 18 q^{40} + 6 q^{41} + 6 q^{43} - 6 q^{44} - 24 q^{46} - 24 q^{47} + 39 q^{50} + 21 q^{52} - 9 q^{58} - 24 q^{59} - 18 q^{61} + 6 q^{62} - 21 q^{64} - 30 q^{65} + 12 q^{67} - 6 q^{68} - 12 q^{71} - 24 q^{73} - 39 q^{74} + 3 q^{76} + 12 q^{79} + 9 q^{80} - 30 q^{82} - 18 q^{83} - 18 q^{85} + 24 q^{86} + 3 q^{88} + 18 q^{89} + 18 q^{92} - 15 q^{94} - 12 q^{95} - 24 q^{97}+O(q^{100}) \) 3 * q + 6 * q^4 - 3 * q^8 - 9 * q^10 - 3 * q^11 + 12 * q^16 - 12 * q^19 - 21 * q^20 + 6 * q^23 + 15 * q^25 + 9 * q^26 - 12 * q^29 + 6 * q^31 - 6 * q^32 + 24 * q^34 - 15 * q^38 - 18 * q^40 + 6 * q^41 + 6 * q^43 - 6 * q^44 - 24 * q^46 - 24 * q^47 + 39 * q^50 + 21 * q^52 - 9 * q^58 - 24 * q^59 - 18 * q^61 + 6 * q^62 - 21 * q^64 - 30 * q^65 + 12 * q^67 - 6 * q^68 - 12 * q^71 - 24 * q^73 - 39 * q^74 + 3 * q^76 + 12 * q^79 + 9 * q^80 - 30 * q^82 - 18 * q^83 - 18 * q^85 + 24 * q^86 + 3 * q^88 + 18 * q^89 + 18 * q^92 - 15 * q^94 - 12 * q^95 - 24 * q^97

Introduction

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